A Bijective Proof of Garsia's Q-Lagrange

8/12/2019 A Bijective Proof of Garsia's Q-Lagrange

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A Bijective Proof of Garsias q-Lagrange

Inversion Theorem

Dan W. SingerTiernan Communications

5751 Copley DriveSan Diego, CA [email protected]

Submitted: March 4, 1997Accepted: April 25, 1998

Abstract

A q-Lagrange inversion theorem due to A. M. Garsia is proved by

means of two sign-reversing, weight-preserving involutions on Catalan

trees.

1 Introduction

LetF(u) be a formal power series withF(0) = 0,F(0)= 0 (delta series). ThenF(u) has an inverse f(u) which satisfies

n=k

F(u)kunf(u)

n =uk

and

n=k

f(u)kunF(u)

n =uk

for all k 1, where |un means extract the coefficient ofun.

The coefficients of f(u)n may be expressed in terms of the coefficients of

F(u) by means of the Lagrange inversion formula

f(u)n|uk = unF(u)

F(u)k+1

u1

.

AMS Subject Classification 05E99 (primary), 05A17 (secondary)Keywords: q-Lagrange inversion, Catalan trees


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the electronic journal of combinatorics 5 (1997), #R26 2

The q-Lagrange inversion problem may be stated as follows: given a deltaseriesF(u) and a sequence of formal power series {Fk(u)}which is a q-analogueof{F(u)k}, find {fk(u)}such that

n=k

Fk(u)|unfn(u) =uk (1.1)

and

n=k

fk(u)|unFn(u) =uk (1.2)

for all k 1. If{fk(u)} satisfies equations (1.1) and (1.2) then fk(u) is a q-analogue off(u)k for each k , where f(u) =F1(u). We say that{F

k(u)} and

{fk(u)} are inverse sequences.

There are several solutions to theq-Lagrange inversion problem appearing inthe literature see for example Andrews [2], Garsia [7], Garsia and Remmel [9],Gessel [10], Gessel and Stanton [11][12], Hofbauer [13], Krattenthaler [15], Singer[17][18]. Singer [17] proved an inversion theorem, based on a generalization ofGarsias operator techniques, which unifies and extends theq-Lagrange inversiontheorems of Garsia [7] and Garsia-Remmel [9]. Garsia, Gessel and Stanton, andSinger have shown that Rogers-Ramanujan type identities may be derived bymeans ofq-Lagrange inversion.

Several authors have given quite distinct bijective proofs ofq-series identities,many of which may be interpreted as statements about partitions see Andrews

[1][3], Bressoud [4], Garsia and Milne [8], Joichi and Stanton [14], Sylvester [19].An exceptional example is Garsia and Milnes proof of the Rogers-Ramanujanidentities [16], making use of the involution principle. Bressoud and Zeilbergergave an alternative, much shorter proof of these identities in [5]. Zeilbergergave aq-Foata proof of the q-Pfaff-Saalschutz identity [20], inspired by Foatasbijective proof of the Pfaff-Saalschutz identity [6].

In view of the fact that so many q-series identities may be derived by meansofq-Lagrange inversion as well as by bijective methods, it is desirable to have acombinatorial interpretation of the inverse relations (1.1) and (1.2).

In this paper we will give a bijective proof, using sign-reversing, q-weightpreserving involutions applied to Catalan trees, of the following q-Lagrangeinversion theorem due to Garsia ([7], Theorem 1.1):

Theorem 1.1. Let F(u) be a delta series with F(0) = 1. Then there is aunique delta seriesf(u) which satisfies

k=1

Fkf(u)f(uq) f(uqk1) = u. (1.3)


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Moreover, {f(u)f(uq) f(uqk1)} and{F(u)F(u/q) F(u/qk1)} are inversesequences, that is

i=k

F(u)F(u/q) F(u/qk1)uif(u)f(uq) f(uq

i1) = uk (1.4)

and

i=k

f(u)f(uq) f(uqk1)uiF(u)F(u/q) F(u/q

i1) = uk (1.5)

for allk 1.

Our proof of Theorem 1.1 is organized as follows. We will assume

F(u) =k=1

Fkuk (1.6)

is given to us with F1 = 1. In Section 2 we will define C, the set of Catalantrees. We will exhibitf(u) in terms ofC, show that it satisfies (1.3), and proveuniqueness. In Section 3 we will make some additional definitions regardingCatalan trees and prove three lemmas we shall require later. We will thengive distinct bijective proofs of equations (1.4) and (1.5), in Sections 4 and 5,respectively.

2 The set of Catalan trees

We exhibit f(u) combinatorially as follows. LetCbe the set of rooted, planartrees whose internal vertices have out-degree 2. We refer toCas the set ofCatalan trees. We denote byCp the set of Catalan trees with p external vertices.We have

C1 = { }

C2 = { }

C3 = { , , }

C4 = { , , , , , , , , , , }

and so on. We denote by |T| the number of external vertices of the tree T.

Observe that for p 2 we have

Cp =

pk=2

{

. . . TkT1 T2

:T1, . . . , T k C, |T1| + + |Tk|= p}.


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For each treeT inCwe will define a scalar-weightws(T), a q-weightwq(T), anda composite weightw(T) = ws(T)wq(T). We then set

f(u) =TC

w(T)u|T|,

and show that f(u) has the desired properties.

The scalar-weight function ws is defined recursively by

ws( ) = 1,

ws(

. . . TkT1 T2

) = Fkws(T1)ws(T2) ws(Tk),

where the coefficientsFk are provided by (1.6). IfT CandVI(T) is the set ofinternal vertices ofT, then clearly

ws(T) =

vVI(T)

(Fd(v)),

whered(v) is the out-degree ofv .

The q-weight function wq is defined recursively by

wq( ) = 1,

wq(

. . . TkT1 T2

) =k

i=1

wq(Ti)q(i1)|Ti|.

We now prove equation (1.3). We have

f(u) =TC

ws(T)wq(T)u|T|

= u +

TC\{ }

ws(T)wq(T)u|T|

= u +k=2

T1,...,TkC

ws(

. . . TkT1 T2

)wq(

. . . TkT1 T2

)u|T1|++|Tk|

= u +

k=2 T1,...,TkC(Fk)

k

i=1ws(Ti)wq(Ti)q

(i1)|Ti|u|Ti|

= u k=2

Fk

ki=1

TC

ws(T)wq(T)q(i1)|T|u|T|

= u k=2

Fk

ki=1

f(uqi1),


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which implies (1.3).

We next show that f(u) is unique. Suppose a(u) is a delta series whichsatisfies

k=1

Fka(u)a(uq) a(uqk1) = u.

We will prove

a(u)|up =TCp

w(T) = f(u)|up

by induction on p.

We have

a(u) = u k=2

Fkf(u)f(uq) f(uqk1),

hence

a(u)|u1 = 1 =w( ).

Assume

a(u)|un =TCn

w(T)

for all 1 n < p. Then

a(u)|up = k=2

Fka(u)a(uq) a(uqk1)

up

= k=2

Fk

e1++ek=p

a(u)|ue1 a(u)|uek q

ki=1(i1)ei

=k=2

e1++ek=p

(Fk)k

i=1

TCei

w(T)q(i1)|T|

=

k=2 e1++ek=p T1Ce1 ...TkCekw(

. . . TkT1 T2

)

=TCp

w(T).

Thereforea(u) = f(u).


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3 A closer look at Catalan trees

For any T Cwe define the crown ofT, C(T), recursively as follows. IfT is aheight 0 or 1 tree then C(T) = T. IfT has height 2, write

T =

. . . TkT1 T2

.

Letr be the least index such thatTr has height 1. We set C(T) = C(Tr).

IfT C\{ }thenC(T) is the height 1 subtree ofTconsisting of the depth-first occurring height-maximal internal vertex ofTand its successors inT. Wewill denote byD(T) the tree derived fromTby replacingC(T) with an externalvertex.

We label the position of the external vertices of any tree T by 1,2, . . . ,

|T| in depth-first order. We denote by P(T) the position of the depth-firstexternal vertex ofC(T) inT. These definitions are illustrated in Figure 3.1 andFigure 3.2.

P ( T ) = 4

C ( T ) =

1

3

4 5

2

6

7

8

9 1 01 1

1 2

1 3 1 4

Figure 3.1: C(T) andP(T)

The statistics P(T) and P(D(T)) are related as follows.

Lemma 3.1. For any treeT in Cwe have

P(T) P(D(T)) + |C(D(T))| 1. (3.1)


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12

3 4 5

6

7 8 9

1 0

1 1 1 2

Figure 3.2: D(T)

Proof. If

T = or T=

. . .1 2 k

then C(T) = T and D(T) = . Therefore P(T) = P(D(T)) =|C(D(T))| = 1,and (3.1) is true in this case.

Now let Tbe a height 2 tree. Write

T =

. . . TkT1 T2

.

Letr be least such that ht(Tr)> 0. Then we may write

T =

1 Tr Tk. . .. . . r - 1

and D(T) =

1 . . . . . . TkD ( T )rr - 1

.

Suppose ht(Tr) = 1. ThenC(T) = C(Tr) = Tr and P(T) = r . There are twocases to consider.

Case 1. ht(D(T)) = 1. In this case we have

C(D(T)) = D(T) =

. . .1 2 k

,


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P(D(T)) = 1, and

P(T) = r k = P(D(T)) + |C(D(T))| 1.

Case 2. ht(D(T))> 1. We are assuming that ht(Tr) = 1, henceD(Tr) = ,and there must be a least index s > r such that ht(Ts) > 0. This impliesC(D(T)) = C(Ts) and P(D(T)) s. Therefore

P(T) = r < s P(D(T)) P(D(T)) + |C(D(T))| 1.

We prove the lemma in general by induction on ht(T), having treated thecase ht(T) 1 above. Assume (3.1) is true for all trees of height a. LetT be

a tree of height a + 1. Write

T =

. . . TkT1 T2

.

Since the root (as does every internal vertex) has out-degree 2, ht(Ti) aforeach i. Let r be least such that ht(Tr)> 0. As before we may write

T =

1 Tr Tk. . .. . . r - 1

and D(T) =

1 . . . . . . TkD ( T )rr - 1

.

We have C(T) = C(Tr) and P(T) = r 1 + P(Tr). We may assume withoutloss of generality that ht(Tr) > 1, having treated the case ht(Tr) = 1 above.This allows us to write ht(D(Tr))> 0, C(D(T)) =C(D(Tr)), and P(D(T)) =r 1 + P(D(Tr)). By the induction hypothesis we have

P(Tr) P(D(Tr)) + |C(D(Tr))| 1.

Therefore

P(T) = r 1 + P(Tr)

r 1 + P(D(Tr)) + |C(D(Tr))| 1

= P(D(T)) + |C(D(T))| 1.

This completes the proof.

LetNbe a height 1 tree. We denote byTaNthe tree obtained by replacingtheath external vertex ofTin depth first order by N. We will need the followingresult.


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Lemma 3.2. With notation as above, let S = T a N, where a P(T) +|C(T)| 1. Then C(S) =N.

Proof. By induction on ht(T). The case ht(T) 1 is trivial. Consider ht(T)>1. Write

T =

. . . TkT1 T2

.

Letr be the least index such that ht(Tr)> 0. Then we may write

T =

1 Tr Tk. . .. . . r - 1

.

ClearlyC(S) =N in case a < r.

Now suppose we have

r a P(T) + |C(T)| 1.

Since the depth-last external vertex ofC(T) occupies position

P(T) + |C(T)| 1

within T, and C(T) is found in Tr, N must be attached to Tr. Moreover,regardingTr as an independent tree, N is attached to Tr at position a r+ 1.WriteSr = Trar+1N. Then we may write

S=

1 S rT

k. . .. . . r - 1

.

We haveP(T) = r 1 + P(Tr) and C(T) = C(Tr), hence

a r+ 1 (P(T) + |C(T)| 1) r+ 1 =P(Tr) + |C(Tr)| 1.

Since ht(Tr) < ht(T), by the induction hypothesis we must have C(Sr) = N.

SinceC(S) = C(Sr), we are done.

We may q-label the external vertices of a tree T with positive integers insuch a way that its q-weight is

wq(T) = qsum of labels inT .


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The labelling is defined by induction on the height of a tree.

Label the height 0 tree by 0. Having labelled the external vertices of all treesof height a or less, we label any height a + 1 tree

T =

. . . TkT1 T2

by increasing every label in Ti byi 1 for eachi. The sum of the labels in T is

ki=1

(i 1)|Ti| + sum of labels inTi.

Set

wL(T) = qsum of labels in T.

We have

wL( ) = 1,

wL(

. . . TkT1 T2

) = q

ki=1(i1)|Ti|+ sum of labels in Ti

=k

i=1

q(i1)|Ti|wL(Ti),

hencewL(T) = wq(T) for all T.

The q-labelling of the tree shown in Figure 3.1 is depicted in Figure 3.3.

An important fact about the q-weight of a tree is recorded in the followinglemma.

Lemma 3.3. For anyT inC, wq(T)andwq(D(T))are related by the equation

wq(T) = wq(D(T))q(P(T)1)(|C(T)|1)+(|C(T)|2 ). (3.2)

Proof. It will suffice to show that the depth-firstq-label on C(T) in T isP(T)1.If this is true then the labels on C(T) are

P(T) 1, (P(T) 1) + 1, . . . , (P(T) 1) + |C(T)| 1,

hence the sum of the labels on C(T) is

(P(T) 1)|C(T)| +

|C(T)|

2

.


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0

1

2

3 4 5

4

3

4 5 6

3

4 5

Figure 3.3: q-Labels

This contribution to the exponent of q in wq(T) is replaced by P(T) 1 inwq(D(T)), since D(T) is obtained from T by replacing the crown by a singleexternal vertex. Hence

sum of labels in T=

sum of labels in D(T) (P(T) 1) + (P(T) 1)|C(T)| + |C(T)|

2 =sum of labels in D(T) + (P(T) 1)(|C(T)| 1) +

|C(T)|

2

.

This implies (3.2).

We show that the depth-first q-label onC(T) inT isP(T) 1 by inductionon|D(T)|. If|D(T)|= 1 then

T = or T =

. . .1 2 k

.

In either case C(T) = T, hence P(T) = 1 and the depth-first label ofC(T) is0 = P(T) 1.

Now assume that for all T such that |D(T)| a, the depth-first q-label onC(T) in T isP(T) 1. LetTbe a tree with |D(T)|= a + 1. Write

T =

. . . TkT1 T2

.


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Since |D(T)|> 1, it must be true that some Ti has height 1. Let r be leastsuch that ht(Tr) 1. Then Tmay be expressed in the form

1 Tr Tk. . .. . . r - 1

.

We have |D(Tr)| a. By the induction hypothesis, the depth-first q-label onC(Tr) inTr (regarded as an independent tree, not a subtree ofT) isP(Tr) 1.By the way q-labelling is defined, the q-labels on Tras a subtree ofT are obtainedby increasing each of the q-labels ofTr as an independent tree by r 1. Hencethe depth-firstq-label onC(Tr) in T isP(Tr)+ r 2. ButP(T) = r 1 + P(Tr),therefore the depth-firstq-label onC(Tr) in T isP(T)1. SinceC(T) = C(Tr),

we are done.

On occasion we will identify Ck, the cartesian product ofk copies of the setC, with the set of all Catalan trees whose root branches out to k subtrees. Inthis context we have

P(T1, . . . , T k) = P(

. . . TkT1 T2

)

and

|(T1, . . . , T k)|= |T1| + + |Tk|.

Given

T =

. . . TkT1 T2

we define the statistic

B(T) =

0 ifT1= = Tk = ,max{i: Ti= } otherwise.

We also set

B(T1, . . . , T k) = B(

. . . TkT1 T2

)

for any (T1, . . . , T k) Ck.


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4 Proof of Equation 1.4

We will prove 1.4 for allk 2, the case k = 1 having been treated in Section 2.We denote byNthe set of height 1 trees (nests). For eachk 2 letRk be theset of composite objects defined by

Rk = {(N, T) : N Nk, T C|N|}.

For each composite ob ject (N, T) Rkwe define a scalar-weight Ws(N, T) and aq-weight Wq(N, T) as follows. Write N= (N1, . . . , N k) and T = (T1, . . . , T |N|).Say|Ni|= ei for each i. We set

Ws(N, T) =k

i=1Fei

|N|

j=1ws(Tj)

and

Wq(N, T) =k

i=1

q(i1)ei wq(T).

We also set

W(N, T) = Ws(N, T)Wq(N, T).

Lemma 4.1. (N,T)Rk

W(N, T)u|T| =

p=k

F(u)F(u/q) F(u/qk1)upf(u)f(uq) f(uq

p1).

Proof. Observe that

W(N, T) =k

i=1Feiq

(i1)ei

|N|

j=1w(Tj)q

(j1)|Tj |.

Hence (N,T)Rk

W(N, T)u|T| =


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p=k e1++ek=p T1,...,TpCk

i=1 Feiq(i1)ei

p

j=1 w(Tj)q(j1)|Tj |u|Tj| =

p=k

e1++ek=p

ki=1

Feiq(i1)ei

pj=1

TC

w(T)q(j1)|T|u|T| =

p=k

F(u)F(u/q) F(u/qk1)upf(u)f(uq) f(uq

p1).

LetNk denote the vector (

k , , . . . , ). ThenW(Nk, Nk) = 1 and |Nk|= k.In order to prove (1.4), we partition the set Rk = Rk\{(Nk, Nk)} into twodisjoint setsR+k andR

k, and exhibit a sign reversing involution on R

k which

maps R+k ontoRk. That is,

2 = identity and

W((N, T)) = W(N, T). (4.1)

This will give us (N,T)Rk

W(N, T)u|T| =W(Nk, Nk)u|Nk| =uk. (4.2)

We then combine this result with Lemma 4.1.

We set

R+k ={(N, T) Rk\{(Nk, Nk)}: B(T) |N| P(N) + 1 and N=Nk}

and

Rk ={(N, T) Rk\{(Nk, Nk)}: B(T)> |N| P(N) + 1 orN=Nk}.

We will actually construct two maps,

+ :R+k Rk and

:Rk R+k,

and show + =+ = identity.

It is helpful to visualize composite objects in Rk as marked trees. Let

(N, T) Rk be given. WriteN = (N1, . . . , N k) and T = (T1, . . . , T |N|). Wemay form a single tree by placing each non-trivial Ti on top of the i

th externalvertex of the tree

. . . Nk

N1

N2

.


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We mark the tree by distinguishing N.

For example, let

N=

, , , , ,

,

T =

, , , , , , , , , ,

.

We identify (N, T) with the marked tree in Figure 4.1.

Figure 4.1: (N, T)

This particular tree lies in R+6: we have|N|= 11,P(N) = 3, andB(T) = 9,therefore

B(T) |N| P(N) + 1.

The maps + and are defined as follows. The reader may wish toconsult Figures 4.34.5 at this point for an illustration of the action of +.Let (N, T) Rk be given. As before, we write N = (N1, . . . , N k) and =(T1, . . . , T |N|). We can regardNas a height 2 tree with a base branching out

tok subtrees. Suppose (N, T) R

+

k. ThenB(T) |N| P(N) + 1 andN=Nk.

We set

+(N, T) = (+1(N, T), +2(N, T)),


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where

+1(N, T) = D(N),

+2(N, T) = (T1, . . . , T |N||C(N)|P(N)+1,

S S1 |C(N)|

. . .

,

P(N)1 , . . . , ),

and

Si= T|N||C(N)|P(N)+1+i for 1 i |C(N)|.

Observe that the depth-last vertex ofC(N) occupies positionP(N)+ |C(N)|1within N, hence P(N) + |C(N)| 1 |N|. Therefore + is well-defined in sofar as|N| |C(N)| P(N) + 1 0.

Lemma 4.2. With notation as above, the composite object +(N, T) lies inRk.

Proof. This is clear ifD(N) = Nk. IfD(N)=Nk, we must verify

B(+2(N, T))>|D(N)| P(D(N)) + 1. (4.3)

By construction we haveB(+2(N, T)) = |N| |C(N)| P(N)+2. We also have|D(N)|=|N| |C(N)| + 1. Hence (4.3) is equivalent to

|N| |C(N)| P(N) + 2> |N| |C(N)| + 1 P(D(N)) + 1,

that is P(D(N))> P(N). But this is clear because the restrictionD(N)=Nkmeans that Nmust be a height 2 tree with at least two nests at height one.The crown ofNis the depth-first nest at height one in N. The crown ofD(N)is the depth-second nest at height one in N. See Figure 4.2.

C(N)

P(N)=2

C(D(N))

P(D(N) )=4

Figure 4.2: NversusD(N)


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We now define . Consider (N, T) Rk. ThenB(T) > |N| P(N) +1 orN=Nk.

Claim: B(T)> 0. This is clear ifN=Nk, for then T =Nk. IfN=Nk, thenwe must have

B(T)> |N| P(N) + 1 1.

Hence we may write

T = (T1, . . . , T B(T)1,

S S1 b

. . .

,

|N|B(T), . . . , )

for someS1, . . . , S b C. We set

(N, T) = (1(N, T), 2(N, T)),

where 1(N, T) is obtained by replacing N|N|B(T)+1 by the nest

1 b. . .

inN, and

2 (N, T) = (T1, . . . , T B(T)1, S1, . . . , S b,

|N|B(T) , . . . , ).

Lemma 4.3. With notation as above, the composite object (N, T) lies inR+k.

Proof. Clearly1(N, T) =Nk. We must verify

B(2(N, T)) |1(N, T)| P(

1(N, T)) + 1. (4.4)

Note that ifN=Nk, then the crown of1(N, T) is the nest

1 b. . .

.

IfN=Nk

, then the condition

B(T)> |N| P(N) + 1

forces

P(N)> |N| B(T) + 1,


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and the crown of 1(N, T) is again the nest

1 b. . .

.

In either case, we have

P(1(N, T)) = |N| B(T) + 1.

We also have

|1(N, T)|=|N| + b 1.

Hence the inequality (4.4) is equivalent to

B(2(N, T)) b + B(T) 1.

This inequality is true becauseSb occupies coordinateb + B(T) 1 in 2(N, T).

In order to see that + = + = identity, it is useful to look at anexample. Consider the composite object (N, T) R+ depicted in Figure 4.1.The action of+ on (N, T) is the following. Write (N, T) as an exploded diagramby separating the non-trivial trees in T from N and recording their relativepositions in depth-first order. See Figure 4.3.

1

549

Figure 4.3: (N, T) exploded

RemoveC(N) fromNand reattach at the external vertex ofD(N) locatedby walking back P(N) positions from the depth-last external vertex. Call thisnew tree I(N). See Figure 4.4. The relocated nest is labelled R.


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1

5

I (N )

4

9

R

Figure 4.4: I(N)

The position ofR on D(N) is

|D(N)| P(N) + 1 =|N| |C(N)| P(N) + 2.

R forms the base for a tree in +2(N, T). To obtain +(N, T) we reattach the

non-trivial trees ofTat the same relative positions on I(N). See Figure 4.5.

Since the depth-last external vertex on R occupies position

(|N| |C(N)| P(N) + 2) + (|C(N)| 1) = |N| P(N) + 1 B(T)

within I(N), none of the non-trivial trees in T clearR. ThereforeR is the basefor the depth-last non-trivial tree sitting on D(N) in +(N, T), and

B(+2(N, T)) = |N| |C(N)| P(N) + 2.

The action of reverses this procedure. The baseR is removed and thenreplaced on D(N) at position

|+1(N, T)| B(+2(N, T)) + 1 =

(|N| |C(N)| + 1) (|N| |C(N)| P(N) + 2) + 1 =P(N).

ThereforeN is recovered. The non-trivial trees among

T1, . . . , T B(T)


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Figure 4.5: +(N, T)

are replaced onN in the same relative positions they occupied on I(N). HenceT is recovered.

The action of in general is to take the base

1 b. . .

of

TB(T)=

S S1 b

. . .

and replace it on Nat position|N| B(T) + 1, forming the crown of1(N, T).The non-trivial trees among

T1, . . . , T B(T)1, S1, . . . , S b

are replaced on 1(N, T) at the same relative positions. + reverses this proce-

dure: I(1(N, T)) is obtained from 1(N, T) by removing

C(1 (N, T)) =1 b

. . .

from1(N, T) and replacing it on D(1(N, T)) at position

|1 (N, T)| |C(1(N, T))| P(

1 (N, T))+ 2 =


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(|N| + b 1) b (|N| B(T) + 1) + 2 =B(T).

Hence Nand the base ofTB(T) are recovered. T is recovered when the non-trivial trees among

T1, . . . , T B(T)1, S1, . . . , S b

are replaced on N.

The next two Lemmas show thatis a sign-reversing, q-weight-preserving in-volution. It will suffice to show that + is sign-reversing and q-weight-preserving.

Lemma 4.4.

Ws(+(N, T)) = Ws(N, T).

Proof. Write N= (N1, . . . , N k) andT= (T1, . . . , T |N|). Recall that

+(N, T) = (+1(N, T), +2(N, T)),

where

+1(N, T) = D(N),

+2(N, T) = (T1, . . . , T |N||C(N)|P(N)+1,

S S1 |C(N)|

. . .

,

P(N)1 , . . . , ),

and

Si= T|N||C(N)|P(N)+1+i for 1 i |C(N)|.

For any S= (S1, . . . , S k) Ck letVkI(S) denote the set of internal vertices

of the trees S1, S2, . . . , S k. Then

Ws(N, T) =

vVk

I (N)

Fd(v)

vVk

I (T)

(Fd(v))

and

Ws(+(N, T)) =

vVk

I (D(N))

Fd(v)

vVkI (

+2 (N,T))

(Fd(v))

,

where d(v) is the out-degree ofv. VkI(D(N)) is obtained from VkI(N) by re-

moving the root of C(N). VkI(+2(N, T)) is obtained from V

kI(T) by adding

the root ofC(N) in its new position in +2(N, T). Hence the factor F|C(N)| inWs(N, T) is replaced byF|C(N)| in Ws(

+(N, T)). ThereforeWs(+(N, T)) =Ws(N, T).


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Lemma 4.5.

Wq(+(N, T)) = Wq(N, T).

Proof. We have

Wq(N, T) = q0|N1|1|N2|(P(N)1)|C(N)|(k1)|Nk|

wq(

T T. . .1 a . . .a + 1 | N |

)

and

Wq(+(N, T)) = q0|N1|1|N2|(P(N)1)1(k1)|Nk|

wq(

T . . . T

T . . . T

b + 2 . . . | N | - | C ( N ) | + 11 b

b + 1 a

),

where a = |N| P(N) + 1 B(T) and b = |N| |C(N)| P(N) + 1. NotethatT1, . . . , T a have the sameq-labels in both

A=

T T. . .1 a . . .a + 1 | N |

and

B =

T . . . T

T . . . T

b + 2 . . . | N | - | C ( N ) | + 11 b

b + 1 a

.

Hence

wq(A) =a

i=1

wq(Ti)q(i1)|Ti|q

|N|i=a+1(i1)

=a

i=1

wq(Ti)q(i1)|Ti|q(

|N|2 )(

|N|P(N)+12 )


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and

wq(B) =a

i=1

wq(Ti)q(i1)|Ti|q

|N||C(N)|+1i=b+2 (i1)

=a

i=1

wq(Ti)q(i1)|Ti|q(

|N||C(N)|+12 )(

|N||C(N)|P(N)+22 ).

Therefore

Wq(N, T)

Wq(+(N, T))=

q(P(N)1)(|C(N)|1)+(|N|2 )(

|N|P(N)+12 )(

|N||C(N)|+12 )+(

|N||C(N)|P(N)+22 ) = 1.

Lemmas 4.4 and 4.5 taken together yield equations (4.1) and (4.2), hence wehave completed the proof of equation (1.4).

5 Proof of Equation 1.5

For eachk 1 letSk be the set of composite objects defined by

Sk = {(T, N) : T Ck andN N|T|},

where as before N is the set of height one trees. For each composite object

(T, N) Sk we define a scalar weight Ws(T, N) and a q-weight Wq(T, N) asfollows. WriteT = (T1, . . . , T k) and N= (N1, . . . , N |T|). Say|Ni|= ei for eachi. We set

Ws(T, N) =k

i=1

ws(Ti)

|T|j=1

Fej

and

Wq(T, N) = wq(T)

|T|j=1

q(j1)ej .

We also set

W(T, N) = Ws(T, N)Wq(T, N).


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Lemma 5.1.

(T,N)Sk

W(T, N)u|N

| =

p=k

f(u)f(uq) f(uqk1)upF(u)F(u/q) F(u/q

p1).

Proof. Observe that

W(T, N) =k

i=1

w(Ti)q(i1)|Ti|

|T|j=1

Fej q(j1)ej .

Hence (T,N)Sk

W(T, N)u|N| =

p=k

(T1,...,Tk)Cp

e1,...,ep1

ki=1

w(Ti)q(i1)|Ti|

pj=1

Fej q(j1)ejuej =

p=k

ki=1

TC

w(T)q(i1)|T|u|T|up

pj=1

r=1

Frq(j1)rur =

p=k

f(u)f(uq) f(uqk1)upF(u)F(u/q) F(u/q

p1).

As before, letNk denote the vector (

k , , . . . , ). ThenW(Nk, Nk) = 1. In

order to prove (1.5), we partition the set Sk\{(Nk, Nk)} into two disjoint setsS+k and S

k and exhibit a sign-reversing involution which maps S

+k ontoS

k .

This will give us

(T,N)Sk W(T, N)u|N| =W(Nk, Nk)u

|Nk| =uk. (5.1)

We then combine this result with Lemma 5.1

We set

S+k ={(T, N) Sk\{(Nk, Nk)}:|T| |C(T)| P(T) + 2> B(N)}


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and

Sk ={(T, N) Sk\{(Nk, Nk)}: |T| |C(T)| P(T) + 2 B(N)}.

We will actually construct two maps,

+ :S+k Sk and

:Sk S+k ,

and show + =+ = identity.

As before it is helpful to visualize composite objects as marked trees. Let(T, N) Sk be given. Write T = (T1, . . . , T k) and N = (N1, . . . , N |T|). We

may form a single tree by placing each non-trivial Ni on top of theith externalvertex of the tree

. . . TkT1 T2

.

We mark the tree by distinguishing T.

For example, let

T =

, , , ,

,

N=

, , , , , , , , , , , , , , ,

We identify (T, N) with marked tree in Figure 5.1. This particular tree lies in

S+5 : we have |T| = 16, P(T) = 3, |C(T)|= 3, and B(N) = 7, therefore

|T| |C(T)| P(T) + 2> B(N).

The maps + and are defined as follows. (The reader may wish toconsult Figures 5.25.4.) Let (T, N) Sk be given. Write T = (T1, . . . , T k) andN = (N1, . . . , N |T|). We can regard T as a tree whose root branches out to k

subtrees. Suppose (T, N) S+k . Then

|T| |C(T)| P(T) + 2> B(N).

We set

+(T, N) = (+1(T, N), +2(T, N)),

where

+1(T, N) = D(T),


26/34


Figure 5.1: (T, N)

+2(T, N) = (

|D(T)| N1, . . . , N B(N), , . . . , C (T), . . . , ),

andC(T) is in position

|T| |C(T)| P(T) + 2.

+ is well-defined because

|D(T)| =|T| |C(T)| + 1 |T| |C(T)| P(T) + 2> B(N).

Lemma 5.2. With notation as above, +(T, N) lies in Sk .

Proof. We need to verify

|D(T)| |C(D(T))| P(D(T))+ 2 B(+2(T, N)). (5.2)

We have

|D(T)|= |T| |C(T)| + 1

and

B(+2(T, N)) = |T| |C(T)| P(T) + 2,

hence (5.2) is equivalent to

P(T) P(D(T)) + |C(D(T))| 1.

But this is true by Lemma 3.1


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Now suppose (T, N) Sk . Then

|T| |C(T)| P(T) + 2 B(N).

We set

(T, N) = (1(T, N), 2(T, N)),

where

1(T, N) = T|T|B(N)+1NB(N)

and

2(T, N) = (N1, . . . ,

omit

NB(N) ,|NB(N)|

, . . . , ).Recall thatTaN is the result of attaching N toTat external vertex a. Notethat

|T| P(T) + |C(T)| 1

implies

B(N) |T| |C(T)| P(T) + 2 1.

Lemma 5.3. With notation as above, (T, N) lies in S+k .

Proof. We must verify

|1(T, N)| |C(1(T, N))| P(

1(T, N)) + 2> B(

2(T, N)). (5.3)

We have

|1(T, N)|= |T| + |NB(N)| 1.

The condition

|T| |C(T)| P(T) + 2 B(N)

forces

|T| B(N) + 1 P(T) + |C(T)| 1.

By Lemma 3.2 this implies

C(1(T, N)) = NB(N).


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Therefore

P(1(T, N)) = |T| B(N) + 1.

Hence (5.3) is equivalent to

(|T| + |NB(N)| 1) |NB(N)| (|T| B(N) + 1) + 2> B(2(T, N)),

that is

B(N)> B(2(T, N)).

But this is true by construction.

In order to see that + = + = identity, it is again useful to havean example in mind. Consider the composite object (T, N) S+k depicted in

Figure 5.1. The action of+ on (T, N) is the following. Write (T, N) as anexploded diagram by separating the non-trivial trees in NfromTand recordingtheir relative positions in depth-first order. See Figure 5.2.

1

2

7

Figure 5.2: (T, N) exploded

RemoveC(T) fromTand reattach as part of+2(T, N) at the external vertex

ofD(T) located by walking back P(T) positions from the depth-last externalvertex. See Figure 5.3. The relocated nest is labelled S.

The position ofSon D(T) is

|D(T)| P(T) + 1 =|T| |C(T)| P(T) + 2> B(N).


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1

2

7

S

Figure 5.3: relocated nest

Hence the non-trivial nests of N may be reattached to D(T) in their samerelative positions, all to the left ofSin depth-first order. The result is+(T, N).See Figure 5.4. Note that we now have

B(+2(T, N)) = |T| |C(T)| P(T) + 2.

The action of reverses this procedure. The nest Sis removed and replaced

onD(T) at position|D(T)| B(+2(T, N)) + 1 =

(|T| |C(T)| + 1) (|T| |C(T)| P(T) + 2) + 1 =P(T).

HenceT is recovered. The non-trivial nests among

N1, . . . , N B(N)

are then replaced in the same relative positions. Hence N is recovered.

The action of in general is to remove NB(N) and replace it on T atposition

|T| B(N) + 1,forming the crown of1(T, N). The remaining non-trivial nests ofNare placedon 1(T, N) in the same relative positions.

+ reverses this procedure: thecrown

C(1(T, N)) = NB(N)


30/34


Figure 5.4: +(T, N)

is removed and replaced as a nest on

D(1(T, N)) = T

at position

|1(T, N)| |C(1(T, N))| P(

1(T, N)) + 2 =

(|T| + |NB(N)| 1) |NB(N)| (|T| B(N) + 1) + 2 =B(N).

The rest of the non-trivial nests ofNare then placed on T in the same relativepositions. Hence (T, N) is recovered.

We now show

W(+(T, N)) = W(T, N). (5.4)

We clearly have

Ws(+(T, N)) = Ws(T, N) (5.5)

because, arguing as in Lemma 4.4, the factor F|C(N)| inWs(T, N) is replaced

byF|C(N)|

inWs

(+(T, N)).

Lemma 5.4. With notation as above, Wq(+(T, N)) = Wq(T, N).


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Proof. We have

Wq(T, N) = wq(T)|T|i=1

q(i1)|Ni|

= wq(T)

B(N)i=1

q(i1)|Ni| q|T|

j=B(N)+1(j1)

= wq(T)

B(N)i=1

q(i1)|Ni| q(|T|2 )+(

B(N)2 )

Now

+(T, N) = (D(T), +2(T, N)),

where

+2(T, N)) = (

|D(T)| N1, . . . , N B(N), , . . . , C (T), . . . , )

andC(T) is in position

|T| |C(T)| P(T) + 2.

Hence

Wq(+(T, N)) =

wq(D(T))

B(N)i=1

q(i1)|Ni|

q|D(T)|

j=B(N)+1(j1) + |T||C(T)|P(T)+21

q(|T||C(T)|P(T)+21)|C(T)|

=wq(D(T))

B(N)i=1

q(i1)|Ni| q(|D(T)|

2 )+(B(N)2 )(|T||C(T)|P(T)+1)(|C(T)|1).

Hence

Wq(T, N)Wq(

+(T, N))=

wq(T)

wq(D(T))q(

|T|2)+(

|D(T)|2 )+(|T||C(T)|P(T)+1)(|C(T)|1) =


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wq(T)

wq(D(T))q(

|T|2)+(

|T||C(T)|+12 )+(|T||C(T)|P(T)+1)(|C(T)|1).

By Lemma 3.3,

wq(T)

wq(D(T))=q(P(T)1)(|C(T)|1)+(

|C(T)|2 ).

Hence

Wq(T, N)

Wq(+(T, N))

=q(|T|2)+(

|T||C(T)|+12 )+(|T||C(T)|)(|C(T)|1)+(

|C(T)|2 ) = 1.

Combining equation (5.5) with Lemma 5.4, we arrive at equation (5.4). Thisyields (5.1) and completes the proof of (1.5).

6 Conclusion

In the course of proving Theorem 1.1 we have discovered some properties ofCatalan trees. While there may be many sign-reversing involutions giving riseto a bijective proof of equations (1.4) and (1.5) when q = 1, it appears thatonly two of these are invariant with respect to the q-labelling of trees. In [17]the author derived the Rogers-Ramanujan type identity

k=0(q2; q4)kqk

2

(q4; q4)k(q; q2)k=

(q4; q8)(q; q2)

,

which has a ready interpretation in terms of partitions. This identity was derivedby replacingqbyq2 andu by q in

k=0

(q; q2)k(1)kq(k2)uk

(q2; q2)k(u; q)k=

(u2q; q4)(u; q)

.

The latter identity was derived as an example ofq-functional composition, andthe left-hand side can be expressed combinatorially in terms of weighted Catalantrees. It would be fruitful to seek bijective proofs for identities such as these forthe information they contain about trees in addition to partitions and Ferrersdiagrams.

References

[1] G. E. Andrews,The Theory of Partitions, Encyclopedia of Mathematics 2,Addison- Wesley Publishing Company, Reading, Massachusetts, 1976.


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[2] G. E. Andrews, Identities in combinatorics, II: A qanalog of the Lagrangeinversion theorem, Proc. Amer. Math. Soc. 53(1975), 240245.

[3] G. E. Andrews, Multiple series Rogers-Ramanujan type identities, PacificJ. Math. 114(1984), 267283.

[4] D. M. Bressoud,Analytic and Combinatorial generalizations of the Rogers-Ramanujan identities, Memoirs Amer. Math. Soc. 227(1980).

[5] D. M. Bressoud and D. Zeilberger, A short Rogers-Ramanujan bijection,Discrete Math. 38(1982), 313315.

[6] D. Foata,Une demonstration combinatoire de lidentite de Pfaff-Saalschutz,C. R. Acad. Sci. Paris, Ser. I, 297(1983), 221224.

[7] A. M. Garsia,A qanalogue of the Lagrange inversion formula, Houston J.

Math. 7 (1981), 205237.

[8] A. M. Garsia and S. C. Milne, A Rogers-Ramanujan Bijection, J. Comb.Theory Ser. A 31(1981), 289339.

[9] A. M. Garsia and J. Remmel,A novel form ofqLagrange inversion,Hous-ton J. Math. 12(1986), 503523.

[10] I. Gessel,A non-commutative generalization andq-analogue of the Lagrangeinversion formula, Trans. Amer. Math. Soc. 257(1980), 455481.

[11] I. M. Gessel and D. Stanton,Applications ofqLagrange inversion to basichypergeometric series,Trans. Amer. Math. Soc. 277(1983), 173201.

[12] I. M. Gessel and D. Stanton, Another family of qLagrange inversion in-version formulas, Rocky Mountain J. Math 16(1986), 373384.

[13] J. Hofbauer, A qanalogue of the Lagrange expansion, Arch. Math. 42(1984), 536544.

[14] J. T. Joichi and D. Stanton,Bijective proofs of basic hypergeometric seriesidentities,Pacific J. Math., 127, no. 1, (1987), 103120.

[15] C. Krattenthaler, Operator methods and Lagrange inversion: a unified ap-proach to Lagrange formulas, Trans. Amer. Math. Soc. 305 (1988), 431-465.

[16] S. Ramanujan and L. J. Rogers, Proof of certain identities in combinatoryanalysis,Proc. London Math. Soc. 19(1919), 211216.

[17] D. Singer, Q-Analogues of Lagrange Inversion, Adv. Math. 155 (1995),no. 1, 99116.

[18] D. Singer, Errata in Q-Analogues of Lagrange Inversion, submitted toAdvances in Mathematics.


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[19] J. J. Sylvester, A constructive theory of partitions, arranged in three actsan interact and exodion, Amer. J. Math. 5 (1882), 251330, or pages 1

83 of The Collected Papers, Vol. 4, Cambridge University Press, London,1912; reprinted by Chelsea, New York, 1974.

[20] D. Zeilberger, A q-Foata proof of theq-Saalschutz identity,Europ. J. Com-bin. 8(1987), 461463.

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